Stressfelder

Stressfelder, or stress fields, are spatially varying tensor fields that describe the internal forces per unit area within a continuous material. In continuum mechanics, the Cauchy stress tensor σ(x) assigns a traction vector t on any oriented surface with unit normal n through t = σ n. The field is typically a function of position and, in dynamic problems, time.

The stress field consists of normal stresses and shear stresses; in a local frame, σ has components such as σxx, σyy, σzz, and shear components like τxy. The eigenvalues of σ give principal stresses, and Mohr’s circle provides a graphical representation. The field obeys equilibrium equations: div σ + b = 0, where b is body force density. In dynamics, inertia terms appear and the equations become more complex.

Determining a stress field requires constitutive relations linking stress to deformation, such as Hooke’s law for linear elastic materials σ = C:ε, where ε is the small-strain tensor. Boundary conditions specify tractions or displacements on surfaces. Compatibility conditions ensure strains derive from a continuous displacement field. Solutions yield σ(x) throughout the body.

Stress fields are central to engineering and physics: structural analysis, design against failure, geophysical modeling of Earth’s interior, and materials science. They can exhibit concentrations and singularities near geometric features like notches or cracks, and around dislocations in crystals, producing long-range stress patterns.

Numerical methods, especially the finite element method, are commonly used to compute stress fields in complex geometries. Analytic solutions exist for simple geometries and loading cases.